Estimating the missing values for the incomplete decision matrix and consistency optimization in emergency management

Applied Mathematical Modelling 40 (2016) 254–267

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Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm

Estimating the missing values for the incomplete decision matrix and consistency optimization in emergency management Daji Ergu a, Gang Kou b, Yi Peng c,⇑, Mingshan Zhang d a

College of Electrical & Information Engineering, Southwest University for Nationalities, Chengdu 610041, China School of Business Administration, Southwestern University of Finance and Economics, Chengdu 610054, China c School of Management and Economics, University of Electronic Science and Technology of China, 610054, China d Institute of Southwest Minorities Study, Southwest University for Nationalities, Chengdu 610041, China b

a r t i c l e

i n f o

Article history: Received 4 November 2013 Received in revised form 14 April 2015 Accepted 22 April 2015 Available online 6 June 2015 Keywords: Geometric mean induced bias matrix Global consistency Incomplete matrix Least absolute error Least square method Missing comparisons

a b s t r a c t Unconventional emergency decision making not only involves intangible and conflicting criteria, but also needs a fast response to the emergency incident under the cases of time pressure and incomplete information. It might be an effective way to make full use of the outlier data of incident information and skip some direct comparisons between alternatives to make a fast emergency decision. Focusing on the missing judgments estimation issue in an incomplete comparison emergency decision matrix, this paper extends the geometric mean induced bias matrix to estimate the missing judgments and improve the consistency ratios at the same time. The least absolute error method and the least square method are used to optimize the revised geometric mean induced bias matrix and find the missing values. A numerical example with incomplete information is used to demonstrate the proposed models. A case of emergency decision making simulation is also conducted to show how the proposed model is applied in practice. The results show the proposed models are not only capable of completing missing values, but also can efficiently improve the matrix consistency at the same time. In addition, the proposed model can aid emergency managers to make a fast response to unconventional emergency in the case of lacking complete information. Ó 2015 Elsevier Inc. All rights reserved.

1. Introduction Over the last few decades, unconventional emergencies increase in frequency and intensity, and unconventional emergency management has increasingly become a hot research topic and an important decision problem in the real world [1–3]. However, it is a difficult task to make an effective unconventional emergency decision making because the multiple influence factors of different and unexpected events usually involve intangible and conflicting criteria, which may lead an unexpected event to different outcomes and evolution directions. Therefore, two of the most popular multi-criteria decision making (MCDM) methods, the analytic hierarchy process (AHP) and analytic network process (ANP), have been extensively applied to the emergency decision making. For example, Levy [4] described the use of ANP to improve flood hazard mitigation in the 1998 Yangtze river floods. Levy and Taji [5] presented a group analytic network process (GANP) to support hazard ⇑ Corresponding author. E-mail address: [email protected] (Y. Peng). http://dx.doi.org/10.1016/j.apm.2015.04.047 0307-904X/Ó 2015 Elsevier Inc. All rights reserved.

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planning and emergency management under incomplete information. Ohta et al. [6] integrated AHP and Geographical Information System (GIS) to improve the geographical accessibility of neurosurgical emergency hospitals for elderly people, where pairwise comparison was used to calculate the weights of four criteria, i.e. availability of hospital beds, the maximum road distance of the shortest routes, the elderly population within a 3-km radius and the median road distance of the shortest routes. Manca and Brambilla [7] proposed a methodology based on the AHP to quantitatively assess the emergency preparedness and response in road tunnels. Ju et al. [8] presented a hybrid fuzzy AHP and 2-tuple fuzzy linguistic approach to evaluate emergency response capacity. In the second stage of their model, pairwise comparison technique was used to determine the weights of evaluation criteria and sub-criteria. Ju et al. [9] combined the Dempster–Shafer theory with AHP and TOPSIS to evaluate and select the suitable emergency alternatives. In the process of emergency management, Cosgrave [10] pointed out that there were three constraints which posed particular problems for emergency managers, i.e. time constraint, limited information on which decisions have to be taken, and decision load constraint from a large number of decisions that emergency managers have to take. In addition, different from traditional decision problems, there are two tough situations the decision makers have to face during the process of making an emergency decision. On the one hand, there are no established rules and principles for the unconventional emergency managers to follow. On the other hand, to effectively control the developing and evolution trend of an unexpected emergency event and reduce the impact it caused, an unconventional emergency decision needs to be taken very quickly and often with limited information, especially in the early stage of the disaster occurrence, an emergency decision must be made quickly using partial or incomplete information [11]. Therefore, when using AHP/ANP to identify and evaluate the critical influence factors for an unexpected event, or assess the emergency planning, emergency preparedness and emergency response alternatives, decision makers need to conduct n(n  1)/2 pairwise comparisons if there are n criteria/alternatives. It could be possible that the pairwise comparisons are inconsistent or incomplete because of the large number, time pressure, lack of the expertise or incomplete information [12]. Consequently, the inconsistency and the incompleteness issues in a pairwise comparison matrix have been paid more attention to over the past few decades, in which a number of methods and models have been proposed to tackle the inconsistency issue (e.g. [13–23]). For the missing comparisons estimation in an incompleteness comparison matrix, Carmone et al. [24] investigated the effect of reduced sets of pairwise comparisons in the AHP by a Monte Carlo simulation. Hu and Tsai [25] proposed a well-known back propagation multi-layer perception to estimate the missing comparisons of incomplete pairwise comparison matrices in the AHP. Fedrizzi and Giove [26] developed a new method to calculate the missing elements of an incomplete matrix of pairwise comparison values for a decision problem by minimizing a measure of global inconsistency. Gomez-Ruiz et al. [27] developed a model based on the Multi-Layer Perceptron (MLP) neural network to complete missing values and improve the matrix consistency at the same time. Bozóki et al. [28] studied the extension of the pairwise comparison matrix to the case when only partial information is available. Ju [29] proposed a new method to solve multiple criteria group decision making problems with incomplete weight information under linguistic environment. As mentioned previously, to control the evolution of emergency events and reduce the impact of emergency events caused, the emergency decision makers must quickly respond to the emergency event and make a fast decision in a short period of time using partial or incomplete information. Therefore, under the scenario of emergency decision making by AHP/ANP, the inconsistency and incomplete issues become even worse. The first objective of this paper is to make full use of the critical influence factors (outlier data) and reduce the numbers of pairwise comparisons judgments so as to quickly respond the emergency incident. To achieve such an objective, on the one hand, the emergency evaluation experts are allowed to fill in the most confident pairwise comparisons and disregard certain pairwise comparisons to deal with the lack of knowledge and incomplete information required to make the judgments. On the other hand, emergency decision makers can ask emergency experts to judge the critical influence factors based on the outlier data collected from different scenarios of happened incidents to save time. In this case, we will obtain a number of incomplete comparison matrices, which leads to the second objective of this paper. The second objective of this paper is to propose a model to quickly estimate the missing comparisons in an incomplete matrix while keeping its consistency. To achieve such an objective, we extend the geometric mean induced bias matrix (for short GMIBM hereinafter) proposed by Ergu et al. [30] to estimate the missing comparisons in an incomplete comparison emergency decision matrix whilst keeping its global consistency. Different from the revised geometric mean (RGM) method proposed by Gomez-Ruiz et al. [27], the adapted GMIBM only requires the original information of the incomplete comparison matrix and is independent of the way of deriving priority weights from a pairwise matrix. Specifically, the missing judgments are first filled in by unknown variables, and then the adapted GMIBM is applied to obtain a revised ‘complete’ pairwise matrix. To keep the global consistency and estimate the missing judgments, the least absolute error (LAE) method and the least square method (LSM) are used to optimize the objective function and find the optimal solution of missing comparisons. The main contributions of this paper are fourfold: (1) We extend the GMIBM to estimate the missing judgments and improve the consistency ratio at the same time by making full use of the original incomplete information. (2) We propose to apply the least absolute error method and the least square method to optimize the revised GMIBM and find the missing values. (3) The theorems of two corollaries in Ergu et al. [30] are proved mathematically for the first time.

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(4) We propose to skip the unimportant assessment factors and focus on the extremely important factors during the process of unconventional emergency decision making, and apply the proposed method to estimate the missing comparisons. The rest of this paper is organized as follows: next section briefly describes the theorem of GMIBM. In Section 3, the theorem of GMIBM for incomplete comparison matrix is extended. The least absolute error and the least square method are also introduced to construct the optimization problems and find the optimal solution for the missing comparisons. Subsequently, a numerical example and a practical emergency decision making simulation are used to demonstrate how the proposed model can be applied in practice in Section 4. We conclude this paper in Section 5.

2. The theorem of geometric mean induced bias matrix (GMIBM) To identify and adjust the inconsistent elements in a pairwise comparison matrix, Ergu et al. [30] proposed a GMIBM model. In this paper, the GMIBM is further extended to estimate the missing judgment in incomplete matrices under such cases that fast emergency decisions are made either by deliberately ignoring some unimportant comparisons or by using the critical influence factors. Note that the missing judgments can also occur because of the time pressure or lack of complete information. Since the revised complete comparison matrices should satisfy the consistency requirement, we first briefly review the related theorems and corollaries of GMIBM for consistency case next. Theorem 1. The GMIBM C should be a U matrix if judgment matrix A is perfectly consistent, that is,

0vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 u n Y u n   A ¼ ðcij Þ ¼ @t aik akj  aji A ¼ U C¼A T

if

aik akj ¼ aij ;

ð1Þ

k¼1

ffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Q  ¼ ða ij Þnn ¼ n nk¼1 aik akj where A represents an n-by-n geometric mean matrix composed of all geometric mean of aik akj for nn 2 3 1  1 . .. . T all i, j and k, U ¼ 4 .. . .. 5, ‘‘n’’ denotes the order of A, A represents the transpose of matrix A. Symbol ‘’denotes Hadamard 1  1 product (e.g. C ¼ A  B means cij ¼ aij bij for all i and j). Theorem 2. The GMIBM C should be a U matrix if judgment matrix A is perfectly consistent, that is,

0vffiffiffiffiffiffiffiffiffiffiffiffi vffiffiffiffiffiffiffiffiffiffiffiffi 1 u n u n Y Y u u n n T t t @ C ¼ LR  A ¼ ðcij Þ ¼ aik  akj  aji A ¼ U k¼1

if

aik akj ¼ aij ;

ð2Þ

k¼1

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  Q where L ¼ n nk¼1 aik ði ¼ 1; . . . ; nÞ, represents an n-by-one column matrix composed of geometric mean of rows in matrix A. n1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qQ R ¼ n nk¼1 akj denotes an n-by-one row matrix composed of geometric mean of columns in matrix A. 1n

Corollary 1. The GMIBM C should be as close as possible to a U matrix if judgment matrix A is approximately consistent. Proof. If the judgment matrix is approximately consistent, i.e. aik akj  aij for all i, j and k. By theorem 2, we have,

vffiffiffiffiffiffiffiffiffiffiffiffi vffiffiffiffiffiffiffiffiffiffiffiffi vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vffiffiffiffiffiffiffiffiffiffiffi u n u n u n u n qffiffiffiffiffi Y Y Y Y u u u u n n n n cij ¼ t aik  t akj  aji ¼ t aik akj  aji  t aij  aji ¼ n anij  aji ¼ aij aji ¼ 1 k¼1

k¼1

k¼1

k¼1

Therefore, the GMIBM C is close to a U matrix if the judgment matrix is approximately consistent.

h

Corollary 2. There must be some inconsistent elements in the GMIBM C deviating far away from one if the judgment matrix A is inconsistent. Proof by contradiction. Assume all entries in matrix C equal to one even if the judgment matrix A is inconsistent, that is, aik akj – aij holds for some i, j and k, but cij ¼ 1 ði; j ¼ 1; . . . ; nÞ; namely

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vffiffiffiffiffiffiffiffiffiffiffiffi vffiffiffiffiffiffiffiffiffiffiffiffi vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u n u n u n Y Y Y u u u n n n cij ¼ t aik  t akj  aji ¼ t aik akj  aji ¼ 1 k¼1

k¼1

k¼1

Since aji = 1/aij, we have

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u n u n Y Y u u n n aij ¼ t aik akj ¼ t ail alj ði; j ¼ 1; . . . nÞ k¼1

l¼1

Similarly, since alk = 1/akl, we can obtain

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u n u n u n Y Y Y u u u n n n aij ¼ t ail alk akl alj ¼ t ail alk  t akl alj l¼1

l¼1

l¼1

We previously assume all entries in matrix C equal to one, thus cik = 1 and ckj = 1. Similar to aij, we can obtain,

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u n u n Y Y u u n n t aik ¼ ail alk and akj ¼ t akl alj : l¼1

l¼1

Therefore, aik akj ¼ aij . The result contradicts the previous assumption that aij –aik akj for some j and k, indicating any row or column of matrix C contains at least one non-one entry. h

3. GMIBM for incomplete matrix 3.1. Incomplete pairwise comparison matrix According to the principle of pairwise comparison technique originated by Thurstone [31] and the theorem of AHP proposed by Saaty [32], if there are n qualitative criteria, all pair-compared results are arranged in a matrix A = (aij)nn, where aij > 0, aij = 1/aji and aij = aikakj for i, j, k = 1, 2, . . ., n, and decision makers need to complete n(n  1)/2 pairwise comparisons. To quantify experts’ judgments, Professor Saaty suggests using the fundamental scale of the absolute numbers 1–9 to represent judgments in the decision matrix A. As such, an incomplete matrix refers to decision makers could not fill in n(n  1)/2 pairwise comparisons because of time pressure, unwillingness to make a direct comparisons between alternatives or being unsure of some of the comparisons [13], and there are one or more pairs of missing entries in matrix A, denoted as [28],

0

1

B B a21 B B A¼B  B . B . @ . an1

a12



1

a23

a32 .. .

1 .. .



an3

   a1n

1

C  C C    a3n C C .. .. C C . . A  1 

where aij represent the given values of pairwise comparisons, ‘’ denotes the missing comparisons. Assume there are p missing comparisons in the upper triangular part of the incomplete matrix A, we can introduce some unknown variables x1, x2, . . ., xp to denote the p missing comparisons, and then calculate and estimate the missing comparisons by mathematics tools and models. Due to the reciprocal property of pairwise comparison matrix, the total number of missing comparisons of the revised ‘complete’ matrix A is 2p, and can be written as,

0

1

B a B 21 B B Aðaij ; XÞ ¼ Aðaij ; x1 ;    ; xp Þ ¼ B 1=x1 B . B . @ . an1

a12

x1

1 a32 .. .

a23 1 .. .

1=xp

an3

   a1n

1

   xp C C C    a3n C C .. .. C C . . A  1

To estimate the optimal values of the p unknown variables in matrix A, in the following, the GMIBM model is further extended and adapted.

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3.2. GMIBM for estimating missing comparisons For an incomplete pairwise comparison matrix A, the missing comparisons should be estimated while keeping the global consistency so as to make a valid decision. Based on the aforementioned Theorems and Corollaries, we can derive the following theorem for estimating the missing comparisons in an incomplete comparison matrix. Theorem 3. The geometric mean induced bias error matrix (GMIBEM) e should be equal (or close) to a zero matrix if judgment matrix A is perfectly (or approximately) consistent, that is,

0vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 u n  Y u ¼ 0 if n T  t @ e ¼ A  A  U ¼ ðcij  1Þ ¼ aik akj  aji  1A  0 if k¼1

or

aik akj ¼ aij ; aik akj  aij ;

0vffiffiffiffiffiffiffiffiffiffiffiffi vffiffiffiffiffiffiffiffiffiffiffiffi 1 u n u n  Y Y u u ¼ 0 if n n e ¼ LR  A  U ¼ ðcij  1Þ ¼ @t aik  t akj  aji  1A  0 if k¼1 k¼1 T

aik akj ¼ aij aik akj  aij

ð3Þ

ð4Þ

where AT and A are the transpose matrix and the average matrix of the revised ‘complete’ matrix A with unknown variables x1 and 1/x1; x2 and 1/x2; etc, respectively. The definitions of L and R are the same as defined in formula (2). To find the optimal values of the unknown variables whilst keeping the global consistency, we can establish an overdetermined system of n2 number of equations, in which there are more equations than unknowns, i.e. eij ðx1 ; x2 ; . . . xp ; aij Þ ¼ 0; i, j = 1, 2,. . ., n. The specific steps of the missing comparisons estimation include: Step 1: Fill in the missing comparisons with unknown variables x1 and 1/x1; x2 and 1/x2; etc. Step 2: Construct the GMIBEM e by the following three sub-steps. Step 2.1: Compute a column vector L and a row vector R (see formula (2)).  ¼ L  R. Step 2.2: Compute the geometric mean matrix by formula A Step 2.3: Compute GMIBEM e. Step 3: Establish an overdetermined system of equations by minimizing all entries in the error matrix e, i.e. let eij ðx1 ; x2 ; . . . xp ; aij Þ ¼ 0, i, j = 1, 2, . . ., n hold. Step 4: Solve the overdetermined system of equations. Step 5: Test the revised comparison matrix A by replacing the missing comparisons with the estimated values. For some complicated overdetermined system of equations, especially those that are generated from incomplete matrices with high orders, sometimes it is difficult to find their explicit solutions. In such case, we can construct the following optimization problems instead of the overdetermined system of equations to find optimal solutions of unknown variables. According to the approximated case in formulas (3) and (4), all absolute errors in the error matrix e should be minimized in order to keep the global consistency, thus we have the following optimization problem,

Min

  eðaij ; xÞ ¼ jeij j ¼ eij ðx1 ; x2 ;    xp ; aij Þ ;

s:t: 1=9 6 x 6 9:

ð5Þ

To create one objective function f(aij, x), the commonly used least absolute errors (LAE) method is introduced to simplify the above optimization problem, i.e., all bias absolute error entries in error matrix e are added together, and we can further obtain the following optimization problem by minimizing the sum of absolute errors (SAE) of error matrix e,

Min f ðaij ; xÞ ¼

n X n X j¼1 i¼1

jeij j ¼

n X n  X   eij x; aij ; j¼1 i¼1

ð6Þ

s:t: 1=9 6 x 6 9; where aij is the given decision judgments, and x = (x1, x2, . . ., xp) is the vector of missing comparison variables, which is subject to the 9-point scale proposed by Saaty [32]. One also can minimize the average absolute error to find the solutions of unknown variables, namely, define the corresponding optimization problem as,

Min f ðaij ; xÞ ¼

n X n n X n X X 1 1 jeij j ¼ jeij ðx; aij Þj; nðn  1Þ j¼1 i¼1 nðn  1Þ j¼1 i¼1

ð7Þ

s:t: 1=9 6 x 6 9: Analogously, we can construct the following optimization problem by minimizing the squares of each error in error matrix e to find the optimal solutions.

D. Ergu et al. / Applied Mathematical Modelling 40 (2016) 254–267

 

 

eðaij ; xÞ ¼ e2ij ¼ eij x1 ; x2 ; . . . xp ; aij

Min

2  ;

259

ð8Þ

s:t: 1=9 6 x 6 9: In practice, the above optimization problem is usually transformed into the following least square optimization problem.

Min f ðaij ; xÞ ¼

n X n n X n   X X ðe2ij Þ ¼ eij ðx; aij Þ2 ; j¼1 i¼1

j¼1 i¼1

ð9Þ

s:t: 1=9 6 x 6 9: The corresponding optimization problem of average least square error can be defined as,

Min f ðaij ; xÞ ¼

n X n n X n  X X    1 1 ðe2ij Þ ¼ eij x; aij 2 ; nðn  1Þ j¼1 i¼1 nðn  1Þ j¼1 i¼1

ð10Þ

s:t: 1=9 6 x 6 9: 4. Experimental simulation In this section, a numerical example is first used to examine the validity and effectiveness of the aforementioned two ways of estimating the missing comparisons. Then a case of unconventional emergency decision is studied to show the application of the proposed method in the real world decision making. 4.1. Numerical example Suppose there are four emergency response alternatives that need to be quickly evaluated in a natural disaster emergency incident, denoted as A1, A2, A3, and A4, respectively. For the lack of related information, limitation of expertise and time pressure, one expert only filled in the values of a12, a14, and a23, the comparisons a13, a24, and a34 are missing, as shown below.

2

1 9



1

5

1

69 6 A¼6 4 5

1 5

1  

1 5

3

7 7 7 5

ð11Þ

1

In this simple example, the missing values can easily be estimated by using the perfect consistency condition aij = aikakj, that is, a13 = a12a23 = 5/9, a24 = a21a14 = 9/5, and a34 = a32a24 = 9/25. In the following, we apply the proposed GMIBEM model into this judgment matrix to demonstrate the implementation process and validate the effectiveness of the proposed optimization model. The details are as follows. Case-1: Method of solving overdetermined system of equations Step 1: The revised pairwise comparison matrix A0 with unknown variables x1, x2, and x3 is

2

1

69 6 A0 ¼ 6 61 4 x1 5

1 9

x1

1 1 5

5 1

1 x2

1 x3

1 5

3

x2 7 7 7: x3 7 5

ð12Þ

1

Step 2: Construct the GMIBEM e by following three sub-steps. Step 2.1: Compute a column vector L and a row vector R.

2

1

69 6 61 L¼A ¼6 6 x1 6 45 0

1 9

x1

1

5

1 5

1

1 x2

1 x3

1 5

3

qffiffiffiffi 4

x1 ; 45

ffiffiffiffiffiffiffiffiffiffiffi p 4 x2 7 7 45x2 ; 7 qffiffiffiffiffiffi 4 x x3 7 7 5x31 ; 7 qffiffiffiffiffiffiffi 1 5 4 5 ; x2 x3

sffiffiffiffiffiffisffiffiffiffiffiffiffiffiffiffiffisffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffi 1 4 5x1 4 x2 x3 4 45 4 R¼ ; x1 45x2 x3 5

ð13Þ

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where

R¼

 qffiffiffiffi 4

qffiffiffiffiffiffiffi 

qffiffiffiffiffiffiffi qffiffiffiffiffiffi

45 x1

4

1 45x2

4

5x1 x3

4

x2 x3 5

and L ¼

 qffiffiffiffi 4

p ffiffiffiffiffiffiffiffiffiffiffi 4 45x2

x1 45

qffiffiffiffiffiffiffi T

qffiffiffiffiffiffi 4

x3 5x1

4

5 x2 x3

 ¼LR Step 2.2: Compute the geometric mean matrix by formula A

2

1 6 6 qffiffiffiffiffiffiffiffiffiffiffi 6 4 2025x2 6 x1 6 ¼6 A 6 qffiffiffiffiffiffi 6 4 9x3 6 x21 6 4 qffiffiffiffiffiffiffiffiffiffi 4

ffi qffiffiffiffiffi 2

qffiffiffiffiffiffiffiffiffiffiffi 4

x1 2025x2

1

qffiffiffiffiffiffiffiffiffiffiffiffi 4

225x1 x2 x3

qffiffiffiffiffiffiffiffiffiffiffiffi

225 x1 x2 x3

x3 225x1 x2

4

qffiffiffiffiffiffiffiffiffi 4

1 9x22 x3

x1 9x3

4

1 qffiffiffiffiffiffiffi 4

25x1 x2 x23

qffiffiffiffiffiffiffiffiffiffi 3 4

x1 x2 x3 225

7 qffiffiffiffiffiffiffiffiffiffiffiffi 7 7 4 2 9x2 x3 7 7 qffiffiffiffiffiffiffi2 7 7: 4 x2 x3 7 25x1 7 7 5 1

ð14Þ

Step 2.3: Compute GMIBEM e.

qffiffiffiffiffi ffi qffiffiffiffiffiffiffiffiffiffi 3 qffiffiffiffiffiffiffiffiffiffiffi 2 x1 4 x1 x2 x3 1 4 x1 0 9 4 2025x  1  1 5  1 x 9x 225 2 1 3 7 6 6 qffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffi 7 qffiffiffiffiffiffiffiffiffiffiffiffi 7 6 1 4 2025x2 4 4 225x1 x2 1 1 2 6 9x2 x3 7 1 0 5 x2 x1 x3 7 69 7: e¼6 qffiffiffiffiffiffiffi2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 6 qffiffiffiffiffiffi x x 4 2 3 6 x 4 9x3  1 5 4 x3  1 1 0 1 7 7 6 1 x2 x3 225x1 x2 25x1 1 7 6 5 4 qffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffi 1 4 225 1 4 25x1 4  1 x  1 0  1 x 2 3 2 2 5 x1 x2 x3 9x x x x 2

2 3

2 3

Step 3: Establish the overdetermined system of equations by

8 qffiffiffiffiffiffiffiffiffiffiffi 1 4 2025x2 > 1¼0 > > 9 x1 > > > q ffiffiffiffiffi ffi > > > x 4 9x3  1 ¼ 0 > 1 > x21 > > > > q ffiffiffiffiffiffiffiffiffiffi > > > > 15 4 x 225 1¼0 < 1 x2 x3 ðIÞ q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > x3 > > 1¼0 5 4 225x > 1 x2 > > > > q ffiffiffiffiffiffiffiffi ffi > > 1 4 > > > x2 9x22 x3  1 ¼ 0 > > > > qffiffiffiffiffiffiffi > > : x3 4 25x21  1 ¼ 0 x x

ð15Þ





eij x1 ; x2 ; . . . ; xp ; aij ¼ 0, i, j = 1, 2, . . ., n.

8 qffiffiffiffiffiffiffiffiffiffiffi x1 > 19 4 2025x 1¼0 > > 2 > > > q ffiffiffiffiffi ffi > > > 1 4 x21 > > 1¼0 > x1 9x3 > > > q ffiffiffiffiffiffiffiffiffiffiffiffi > > > > 1 4 225x1 x2 ¼ 0 < x3 5 ðIIÞ qffiffiffiffiffiffiffiffiffiffi > x2 x3 > > 5 4 x1225 1¼0 > > > > > q ffiffiffiffiffiffiffiffiffiffiffiffi > > > 1 4 2 > > x2 9x2 x3 ¼ 0 > > > > qffiffiffiffiffiffiffi > > : 1 4 x2 x23  1 ¼ 0 x3 25x1

2 3

ð16Þ

where ‘I’ represents the system of equations composed of the lower triangular parts of matrix e, while ‘II’ denotes the system of equations composed of the upper triangular parts of matrix e. Step 4: Solve the overdetermined system of equations (I) and (II), we obtain,

x1 ¼

5 ; 9

x2 ¼

9 ; 5

x3 ¼

9 25

The calculated results are the same as the values derived from the perfect consistency condition. Step 5: Test the revised comparison matrix A by replacing the missing comparisons with the estimated values. Since the maximum eigenvalue is kmax ¼ 4; thus CR = 0. If one wants to use the 9-point scale integer number, then the integer value that is closest to 9-point scale can be selected as the optimal value of unknown variables. For instance, x1 ¼ 5=9 ¼ 0:5556  1=2, x2 ¼ 9=5 ¼ 1:8  2 and x3 ¼ 9=25 ¼ 0:36  1=3, replace the unknown variables with these values and test the consistency, the maximum eigenvalue is kmax ¼ 4:0055, and the corresponding consistency ratio CR = 0.0021, which is far less than the consistency ratio threshold 0.1. Case-2: Methods of least absolute error and least square method optimization In addition to solving the overdetermined system of equations, we can find the optimal solutions by using optimization formulas (6) and (9). In the following, both optimization methods are applied to the above numerical example. According to the error matrix e and optimization formula (6), we can construct the least absolute error objective function f1(x) by summing up all absolute entries of error matrix ,

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  sffiffiffiffiffiffiffiffi   sffiffiffiffiffiffiffiffiffiffiffiffiffiffi   rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  sffiffiffiffiffiffiffiffiffiffiffiffi   sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi    1 4 225    1 4 2025x   4 1 x3   4 9x3     4   2   1 þ x1  1 f 1 ðxÞ ¼  þ þ 5  1 þ x  1  1      2       5 x1 x 2 x3    9 x1 225x1 x2 x21 9x22 x3   sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  sffiffiffiffiffiffiffiffi   rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  sffiffiffiffiffiffiffiffiffiffiffi      rffiffiffiffiffiffiffiffiffiffiffiffiffiffi   qffiffiffiffiffiffiffiffiffiffiffiffi     1 4 225x1 x2   4 x1 x2 x3   1 4 x21  1 4  x1   4  4 25x1 þ þ 5 þ þ  x3 9x22 x3   1  1 þ 9 þ  1  1    5  x   x     2025x2 9x x 225 x2 x23 3 3 2   1   sffiffiffiffiffiffiffiffiffiffiffi     1 4 x2 x23 þ   1  x3 25x1

ð17Þ

By least absolute error (LAE thereafter) optimization formula (6), the objective function f1(x) should be minimized to keep the global consistency, and the variables are subject to the 9-point scale, that is, we can construct the following optimization model,

Min f 1 ðxÞ 8 > < 1=9 6 x1 6 9 s:t:

> :

ð18Þ

1=9 6 x2 6 9 1=9 6 x3 6 9

Apply the nonlinear constrained optimization function fmincon in Matlab software to solve this optimization problem, we can get the final solutions of unknown variables x, y and z and the aggregation values of objective function f1(x) after 35 iterations of the algorithm, as shown in Table 1. To display the whole iteration steps of optimization, we plotted the objective function’s value and the final estimated values of unknown variables. Fig. 1(b) shows that the decrease of objective function value is close to zero after the 9th iteration, in which the function value in the 9th iteration is 0.0751224 as listed in Table 1. The final function value, 8.06264  106, is far close to zero and the values of estimated unknown variables plotted in Fig. 1(a) are the same as the results obtained from case 1, indicating the revised complete matrix by the estimated values can be regarded as a perfect consistent matrix. Similarly, if we change the absolute value error of f1(x) to the least square value, we can get the following objective function of least square optimization f2(x).

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffi !2 !2 !2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 !2 1 4 225 x3 1 4 2025x2 4 9x3 4 4  1 þ x1  1 f 2 ðxÞ ¼ þ þ 5  1 þ x  1  1 2 5 x1 x 2 x3 x1 225x1 x2 x21 9x22 x3 0 sffiffiffiffiffiffiffiffi 12 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffi !2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2

x1 1 4 x21 1 4 225x1 x2 4 25x1 4 x1 x2 x3 4 @ A  1  1 þ x3 þ 9 þ þ þ 5  1  1 x1 9x3 5 2025x2 x3 225 x2 x23 1 9



1 þ x2

0 sffiffiffiffiffiffiffiffiffiffiffi 12 qffiffiffiffiffiffiffiffiffiffiffiffi 2 1 4 x2 x23 4 2 @  1A : 9x2 x3 þ x3 25x1

ð19Þ

The least square method (LSM) optimization problem becomes,

Table 1 The Iteration and changes of objective functions and estimated values of variables by LAE and LSM. LAE

LSM

Iter

f1(x)

Iter

f1(x)

Iter

f1(x)

Iter

f1(x)

Iter

f2(x)

0 1 2 3 4 5 6 7 8 9 x y z

5.72734 1.11641 0.745779 0.635837 0.627911 0.348826 0.241981 0.163811 0.152278 0.0751224 0.5556 1.8 0.36

10 11 12 13 14 15 16 17 18 19

0.0460101 0.032345 0.0232039 0.0183333 0.0066731 0.00607772 0.00319724 0.0027623 0.00186187 0.00156795

20 21 22 23 24 25 26 27 28 29

0.000770448 0.000737759 0.000578762 0.000317378 0.00029903 0.000204819 0.000193 0.000129697 4.43606e005 3.37231e005

30 31 32 33 34 35

2.71981e005 2.42419e005 1.33939e005 1.11378e005 9.52586e006 8.06264e006

0 1 2 3 4 5 6 7 8 9 x y z

3.71503 0.175477 0.0838976 0.0741309 0.0383508 0.00944636 0.0011475 5.13844e005 2.56853e006 5.5128e008 0.5556 1.7998 0.36

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Current Point

Current point

2 1.5 1 0.5 0

1

2

3

(a) Number of variables: 3 Current Function Value: 8.06264e-006

Function value

6 4 2 0

0

5

10

15

20

25

30

35

(b) Iteration Fig. 1. Changes of variables and function values by LAE method.

Min f 2 ðxÞ 8 > < 1=9 6 x1 6 9 1=9 6 x2 6 9 s:t: > : 1=9 6 x3 6 9

ð20Þ

To compare with the LAE method, the optimization function fmincon is again applied to solve above optimization problem, the detailed iterations and changes of f2(x) are also listed in Table 1 and plotted in Fig. 2. Fig. 2(b) shows that the function value decreases drastically after the first iteration, and it almost reaches zero after the second iteration. Optimization stopped in the 9th iteration since the predicted change in the objective function, 5.5128  108, is less than the given threshold options, 106. The values of estimated unknown variables showed in Table 1 and plotted in Fig. 2(a) are almost the same as the results obtained from case 1. Therefore, the revised complete matrix by these estimated values satisfies perfect consistency condition.

Current Point

Current point

2 1.5 1 0.5 0

1

2

3

(a) Number of variables: 3 Current Function Value: 5.5128e-008

Function value

4 3 2 1 0

0

1

2

3

4

5

(b) Iteration

6

7

8

Fig. 2. Changes of variables and function values by LSM method.

9

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263

4.2. A case of emergency decision making simulation We have previously demonstrated the implementation of the proposed method by a 44 incomplete matrix. In the following, we apply the proposed method to a case of emergency decision making with incomplete decision information in the real world. During the process of making decision for unconventional emergency, quick assessment and scenario-response are extremely important to save lives and reduce the property losses. Take the Yushu earthquake analyzed by Liu et al. [33] as an example, assume an emergency manager has to quickly evaluate the eight effectiveness assessment indicators proposed by Liu et al. [34] for different scenarios using pairwise comparisons technique. They are: Casualties, Personnel rescued, Rescue workers input, Materials input, Direct economic losses, Indirect economic losses, Effectiveness of the physical environment, Social benefits, denoted as C1, C2, C3, C4, C5, C6, C7 and C8, respectively. According to the pairwise comparisons technique, the manager needs to ask the emergency experts to fill in n(n  1)/2 = 28 numbers of pairwise comparisons. To quickly evaluate these criteria and make a fast decision, the emergency manager decided to reduce the number of comparisons by allowing the surveyed emergency experts to fill in parts of the comparisons, i.e., filling in the most confident comparisons using their experience and expertise knowledge. Assume the emergency manager collected the following incomplete matrix A with 12 missing comparisons from one of the surveyed experts, where missing comparisons are denoted by ‘’. To estimate the missing comparisons and evaluate these criteria, we apply the proposed model to this matrix.

2

1

5

3

7

6

6

1=3 1=4

3

6 1=5 1  5  3  1=7 7 7 6 7 6 6 1=3  1  3  6  7 7 6 6 1=7 1=5  1  1=4  1=8 7 7 6 A¼6 7 7 6 1=6  1=3  1  1=5  7 6 7 6 1=6 1=3  4  1  1=6 7 6 7 6 4 3  1=6  5  1  5 4

7



8

6





ð21Þ

1

First, the missing comparisons are replaced by twelve unknown variables, xi (i = 1, 2, . . ., 12), we can obtain the revised ‘complete’ matrix with variables, denoted as A(x),

2

1

5

6 1=5 1 6 6 6 1=3 1=x1 6 6 1=7 1=5 6 AðxÞ ¼ 6 6 1=6 1=x2 6 6 1=6 1=3 6 6 4 3 1=x3 4

7

3

7

6

6

1=3

x1

5

1

x4

x2

3

x3

3

x5

6

1=x4 1=3

1 1=x7

x7 1

1=4 x9

x8 1=5

1=x5

4

1=x9

1

x11

1=6

1=x8

5

1=x11

1

1=x6

8

1=x10

6

1=x12

1=4

3

1=7 7 7 7 x6 7 7 1=8 7 7 7 x10 7 7 1=6 7 7 7 x12 5

ð22Þ

1

To construct and demonstrate the proposed optimization models, both LAE and LSM are used in the following. Case 1: least absolute error (LAE) method According to formula (6) of LAE, the optimization problem is

 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 8  8 X 8 8 X 8 uY X X  8 t ;   a Min f ðaij ; xÞ ¼ jeij j ¼ a ðxÞa ðxÞ ðxÞ  1 ji ik kj    j¼1 i¼1 j¼1 i¼1  k¼1

ð23Þ

s:t: 1=9 6 x 6 9: Or

 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 8  8 X 8 8 X 8 uY  1 X 1 X 8 t    a Min f ðaij ; xÞ ¼ jeij j ¼ a ðxÞa ðxÞ ðxÞ  1 ji ik kj  8  7 j¼1 i¼1 56 j¼1 i¼1  k¼1 

ð24Þ

s:t: 1=9 6 x 6 9 where aij(x) is the ith row and jth column entry of the revised ‘complete’ matrix A(x), and x=(x1, x2, . . ., x12) is the vector of unknown variables, which is subject to 1/9 to 9 in terms of the 9-point scale proposed by Saaty [32]. Apply the nonlinear constrained optimization function fmincon in Matlab software to solve the optimization problem (23), the detailed values of objective function of each iteration optimization and the final estimated optimal values of variables by LAE are shown in Table 2 and plotted in Fig. 3.

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Table 2 The results of optimization by least absolute error (LAE) method. Iter

f2(x)

Iter

f2(x)

Iter

f2(x)

Iter

f2(x)

Iter

f2(x)

Variables xi

Estimated values

9-Point scale

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

49.8101 34.9387 33.8639 33.0633 33.0178 30.6521 30.4364 28.7696 27.8874 27.2627 27.0684 26.709 26.3971 25.7919 25.6805 25.5275

16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

25.2462 24.9166 24.7736 24.5932 24.324 24.3057 24.2053 24.037 23.8869 23.8756 23.7949 23.7943 23.7549 23.7434 23.7145 23.5934

32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

23.4935 23.4874 23.4758 23.4627 23.4339 23.4188 23.4065 23.4044 23.3964 23.3963 23.3905 23.3858 23.3806 23.3757 23.3734 23.3686

48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63

23.3662 23.3659 23.3654 23.365 23.3649 23.3647 23.3647 23.3638 23.3633 23.363 23.3627 23.3621 23.3618 23.3617 23.3616 23.3613

64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79

23.3611 23.3611 23.361 23.3608 23.3607 23.3607 23.3606 23.3606 23.3605 23.3605 23.3605 23.3605 23.3605 23.3604 23.3604 23.3604

x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12

0.3807 1.8279 0.4735 9.0000 4.2863 0.5264 0.5320 0.1378 0.8928 0.1111 0.2902 0.4232

1/3 2 1/2 9 4 1/2 1/2 1/7 1 1/9 1/3 1/2

8

10

Current Point

10 Current point

8 6 4 2 0

1

2

3

4

5

6

7

9

(a) Number of variables: 12 Current Function Value: 23.3604

11

12

Function value

50 40 30 20

0

10

20

30

40

(b) Iteration

50

60

70

80

Fig. 3. The values of unknown variables and the changing function value by LAE.

Fig. 3 shows the changes of objective function’s value and the final estimated values of unknown variables during the whole iteration steps, in which we can observe that a significant decrease of objective function happens in the 8th iteration, as shown in Fig. 3(b). After 24th iteration, the decrease of objective function becomes unobvious, i.e. the decease only happens at the decimal fraction part of the function value. Finally, the optimization terminated in the 80th iteration. The value of final minimized objective function is 23.3604, and the corresponding average absolute error of each entry by formula (24) is 23.3604/56 = 0.41715. The optimal estimated values of unknown variables showed in Fig. 3(a) are xi = (0.3807, 1.8279, 0.4735,9, 4.2863, 0.5264, 0.5320, 0.1378, 0.8928, 0.1111, 0.2902, 0.4232), i = 1, 2, . . ., 12. Replace the missing comparisons in matrix A(x) with the above optimal values, we can get the revised complete matrix A(x⁄),

2

1

5

3

7

6

6

6 1=5 1 0:3807 5 1:8278 3 6 6 6 1=3 1=0:3807 1 9 3 4:2864 6 6 1=7 1=5 1=9 1 0:532 1=4 6 Aðx Þ ¼ 6 6 1=6 1=1:8278 1=3 1=0:532 1 0:8928 6 6 1=6 1=3 1=4:2864 4 1=0:8928 1 6 6 4 3 1=0:4735 1=6 1=0:1378 5 1=0:2902 4

7

1=0:5264

8

1=0:1111

6

1=3 0:4735 6 0:1378 1=5 0:2902 1 1=0:4232

1=4

3

1=7 7 7 7 0:5264 7 7 1=8 7 7 7 0:1111 7 7 1=6 7 7 7 0:4232 5 1

ð25Þ

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Calculate the maximum eigenvalue of the revised matrix, we can obtain kLAE max ¼ 9:3024. The corresponding right eigenvector is calculated and listed in Table 4 in order to compare it with the results obtained by LSM. If one wants to use the 9-point integer scale proposed by Saaty, it is recommended to choose the closest value to the 9-point integer scale, as shown in the right column in Table 2, the corresponding maximum eigenvalue of these estimated integer values is kLAE max ¼ 9:3125 . Case 2: Least square method (LSM) In addition to LAE method, the LSM could also be used to estimate the optimal values of missing comparisons by minimizing the sum of errors squares of formula (4). The corresponding optimization problem of above example can be constructed as,

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0v 12 u 8 8 X 8 8 X 8 Y X X u 8 2 t @ Min f ðaij ; xÞ ¼ ðeij Þ ¼ aik ðxÞakj ðxÞ  aji ðxÞ  1A ; j¼1 i¼1

j¼1 i¼1

ð26Þ

k¼1

s:t: 1=9 6 x 6 9: Or

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0v 12 u 8 8 X 8 8 X 8 u Y X 1 X 1 8 2 t @ Min f ðaij ; xÞ ¼ ðe Þ ¼ aik ðxÞakj ðxÞ  aji ðxÞ  1A ; 8  7 j¼1 i¼1 ij 56 j¼1 i¼1 k¼1

ð27Þ

s:t: 1=9 6 x 6 9: Similar to LAE method, we apply the nonlinear constrained optimization function fmincon in Matlab software to solve optimization problem (26), the detailed values of objective function of each iteration optimization and the final estimated optimal values can be obtained, as shown in Table 3, and are plotted in Fig. 4. It can be seen from Fig. 4(b) that a significant decrease of objective function happens in the 5th iteration. After 19th iteration, the decrease of objective function only happens on the decimal fraction part of the function values, and the value of final minimized objective function is 33.8381. By formula (27), the corresponding average least square error is 33.8381/56 = 0.6043. Fig. 4(a) also shows the dominated variables are x4 and x5. It only took 4.818181 s to estimate the 12 missing variables on a 2.5 GHz Pentium laptop. The speed could be improved if high configuration hardware is employed. Replace the missing comparisons in matrix A(x) with the estimated optimal values showed on the most right column in Table 3, the revised complete matrix A(x⁄) is,

2

1

5

3

7

6

6

6 1=5 1 0:2633 5 1:6536 3 6 6 6 1=3 1=0:2633 1 9 3 6:1232 6 6 1=7 1=5 1=9 1 0:5027 1=4 6 Aðx Þ ¼ 6 6 1=6 1=1:6536 1=3 1=0:5027 1 0:9743 6 6 1=6 1=3 1=6:1232 4 1=0:9743 1 6 6 4 3 1=0:5038 1=6 1=0:1531 5 1=0:3127 4

7

1=0:7461

8

1=0:1186

6

1=3

1=4

3

1=7 7 7 7 0:7461 7 7 1=8 7 7 7 0:1186 7 7 1=6 7 7 7 0:3896 5

0:5038 6 0:1531 1=5 0:3127 1 1=0:3896

ð28Þ

1

Calculate the maximum eigenvalue and the corresponding right eigenvectors, we have kLSM max ¼ 9:3177, the eigenvectors are listed in Table 4. To compare and analyze the results obtained by LSM and LAE methods, Table 4 summarizes five indicators, including the estimated values, the final weights of 8 factors, the final ranking, the corresponding maximum eigenvalues and the consistency ratios. It can be seen from Table 4 that the estimated value of unknown variable x4 obtained by LAE is the same as the Table 3 The results of optimization by least square method (LSM). Iter

f(x)

Iter

f1(x)

Iter

f(x)

Variables xi

Estimated values

0 1 2 3 4 5 6 7 8 9 10 11

89.0872 61.1581 49.8491 45.1791 41.6983 40.6296 40.0452 39.7068 39.1916 38.52 38.1084 37.7068

12 13 14 15 16 17 18 19 20 21 22 23

37.6695 37.6584 37.0257 36.2223 35.7347 34.8109 34.3041 33.9858 33.8919 33.8521 33.8393 33.8384

24 25 26 27 28

33.8383 33.8382 33.8381 33.8381 33.8381

x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12

0.2633 1.6536 0.5038 9.0000 6.1232 0.7461 0.5027 0.1531 0.9743 0.1186 0.3127 0.3896

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Current Point

Current point

10

5

0

1

2

3

4

5

6

7

8

9

(a) Number of variables: 12 Current Function Value: 33.8381

10

11

12

Function value

100 80 60 40 20

0

5

10

15

20

(b) Iteration

25

30

Fig. 4. Unknown variables and the changing function value of objective function by LSM. Table 4 The comparison of results calculated by LSM and LAE methods. Unknown variables

x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12

Estimated values LAE

LSM

0.3807 1.8279 0.4735 9.0000 4.2863 0.5264 0.5320 0.1378 0.8928 0.1111 0.2902 0.4232

0.2633 1.6536 0.5038 9.0000 6.1232 0.7461 0.5027 0.1531 0.9743 0.1186 0.3127 0.3896

Criteria

Weights (xi) LSM

LAE

LSM

LAE

LAE

LSM

LAE

LSM

C1 C2 C3 C4 C5 C6 C7 C8

0.1890 0.0587 0.2054 0.0177 0.0316 0.0363 0.1527 0.3086

0.1922 0.0555 0.2229 0.0178 0.0329 0.0346 0.1480 0.2961

3 5 2 8 7 6 4 1

3 5 2 8 7 6 4 1

9.3024

9.3177

0.0309

0.0324

Ranking

Maximum eigenvalue (kmax)

Consistency ratio (CR)

one obtained by LSM. In addition, although most of the other estimated values obtained by both methods are slightly different, their integer approximated values within 9-point scale are equal to each other. Take the estimated values of unknown variables x2, x3, x7 and x10 obtained by LAE and LSM methods as examples, the corresponding values and integer approximated values are 1:8279  2, 1:6536  2; 0:4735  1=2, 0:5038  1=2; 0:5320  1=2, 0:5027  1=2; 0:1111  1=9, 0:1186  1=9. Applying the estimated values by both methods, we calculated the corresponding weights of eight criterions, the results show that the final rankings are equal to each other. Besides, the maximum eigenvalue (9.3024) and CR (0.0309) obtained by LAE method are slightly smaller than the maximum eigenvalue (9.3177) and CR (0.0324) obtained by LSM method, showing both consistence ratios are less than the consistency ratio threshold, 0.1. Therefore, the estimated values obtained by both methods satisfy the global consistency conditions and have same rankings. 5. Conclusions In this paper, the GMIBM is adapted and extended to estimate the missing comparisons and improve the consistency ratio at the same time. Specifically, the missing comparisons are first filled in by some unknown variables to obtain a revised ‘complete’ matrix, then construct geometric mean induced bias error matrix by the proposed model. Subsequently, two methods are provided to find the solution of unknown variables: (1) construct an overdetermined system of equations to solve the solution of variables; (2) construct an optimization problem either by the least absolute error (LAE) or by the least square method (LSM) to find the solutions. Different from the existing models for estimating the missing comparisons, our model only requires the original information of incomplete comparison matrix, and is independent of the weights. The correctnesses of two Corollaries of GMIBM are proved mathematically. One 4  4 incomplete matrix and an 8  8 high order incomplete matrix of emergency assessment are used to demonstrate the proposed models. The results show that the proposed models are not only capable of completing missing values, but also can efficiently improve the matrix consistency at the same time.

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Although the proposed model can effectively estimate the missing comparisons in the simulated incomplete comparison emergency decision matrices, which may be one of the effective ways to make a fast emergency decision making by ignoring some unimportant pairwise comparisons or the comparisons beyond the decision maker’s capability, it remains to be validated by more real world emergency fast decision case studies that how fast the proposed model could improve the speed of response in real implication. In addition, it is noted that the limitation of the proposed model is that it can only be used in such case that the pairwise comparison technique is used to collect the experts’ judgments, especially when the AHP/ANP is used in the emergency management. Besides, the validity of the partial judgments provided by emergency experts could impact on the speed and effectiveness of an emergency response by the proposed method in practice. 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